Anisotropic random walks and asymptotically one-dimensional diffusion on abc-gaskets

Journal of Statistical Physics

Volumn 88, Issue 1-2, 1997, Pages 105-128

  Hattori, Tetsuya ^a   Watanabe, Hiroshi ^b  

^a RIKKYO UNIVERSITY   (Japan)

^b NIPPON MEDICAL SCHOOL   (Japan)

Author keywords

Branching process; Diffusion process; Finitely ramified fractal; Random walk; Renormalization group

Indexed keywords

EID: 0031184715     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02508466     Document Type: Article

References (18)

1. M. T. Barlow, Random walks, electrical resistance, and nested fractals, in K. D. Elworthy and N. Ikeda (eds.), Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, Pitman, 1963, pp. 131-157.
2. M. T. Barlow, K. Hattori, T. Hattori, and H. Watanabe, Restoration of isotropy on fractals, Physical Review Letters 75:3042-3045 (1995).
3. M. T. Barlow, K. Hattori, T. Hattori, and H. Watanabe, Weak homogenization of anisotropic diffusion on pre-Sierpiński carpet, in press.
4. M. T. Barlow and E. A. Perkins, Brownian Motion on the Sierpiński Gasket, Probab. Theory Relat. Fields 69:543-623 (1988).
5. B. M. Hambly and T. Kumagai, Heat kernel estimates and homogenization for asymptotically lower dimensional processes on some nested fractals, preprint.
6. M. Fukushima, Dirichlet forms, diffusion processes and spectral dimensions for nested fractals, in Albeverio, Fenstad, Holden, and Lindstrøm (eds.), Ideas and Methods in Mathematical Analysis, Stochastic, and Applications, In Memory of R. Høegh-Krolm, Vol. 1, Cambridge Univ. Press, Cambridge, 1992, pp. 151-161.
7. T. Hattori, Asymptotically one-dimensional diffusions on scale-irregular gaskets, preprint.
8. K. Hattori, T. Hattori, and H. Watanabe, Gaussian field theories on general networks and the spectral dimensions, Progr. Theor. Phys. Supplement 92:108-143 (1987).
9. K. Hattori, T. Hattori, and H. Watanabe, Asymptotically one-dimensional diffusions on the Sierpiński gasket and the abc-gaskets, Probability Theory and Related Fields 100:85-116 (1994).
10. T. Hattori and H. Watanabe, On a limit theorem for non-stationary branching processes, in E. Çinlar et al. (eds.), Seminar on Stochastic Processes, 1992, Birkhäuser, Boston, 1993, 173-187.
11. O. D. Jones, On the convergence of multi-type branching processes with varying environments, preprint.
12. J. Kigami, Effective registances for harmonic structures on p.c.f. self-similar sets, Math. Proc. Camb. Phil. Soc. 115:291-303 (1994).
13. J. Kigami, Harmonic calculus on limits of networks and its application to dendrites, J. Funct. Anal. 128:48-86 (1995).
14. S. M. Kozlov, Harmonization and homogenization on fractals, Commun. Math. Phys. 153:339-357 (1993).
15. S. Kusuoka, A diffusion process on a fractal, in K. Ito and N. Ikeda (eds.), Probabilistic methods on mathematical physics, Proc. Taniguchi Symp. Kinokuniya, Tokyo, 1987, pp. 251-274.
16. S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. RIMS. 25:659-680 (1989).
17. T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 420:1-128 (1990).
18. Z. Vondraček, A characterization of certain diffusions on the Sierpiński gasket by the exit distributions, Journal of Theoretical Probability 9:335-352 (1996).